beyondbinaryfandomcom-20200215-history
Almost Perfect Numbers
Almost Perfect Numbers (APNs) are defined on this wiki as numbers expressible in the form of a product of an Almost Mersenne Prime and the next smallest Power of 2 (e.g. 15 & 8 => 120, 63 & 32 = 2016). Perfect Numbers (PNs) are formed by taking the product of the Mersenne Primes and their next smallest Power of 2 (e.g. 3 & 2 = 6, 7 & 4 = 28). Mersenne Primes A Mersenne prime is a number of the form: : 2^n -1 ::for n=1,2,3,... :that are also prime e.g. : 2^2-1 = 3 : 2^3-1 = 7 : 2^5-1 = 31 : 2^7-1 = 127 : 2^{13}-1 = 8,191 Almost Mersenne Primes An 'almost Mersenne prime' (aMp) can be defined as a number of the form: : 2^n-1 :that is not a prime number (equiv.: 'is an almost prime number') e.g. : 2^0-1 = 0 : 2^1-1 = 1 : 2^4-1 = 15 : 2^6-1 = 63 : 2^8-1 = 255 : 2^9-1 = 511 : 2^{10}-1 = 1,023 : 2^{11}-1 = 2,047 : 2^{12}-1 = 4,095 Alternate definitions of Almost Perfectness Halves and Doubles of Perfect Numbers Such products are perfect only when the exponent of the power of 2 is one less than the exponent of the next power of 2 above the Mersenne prime. Any other power of 2 produces a number that is harmonic with the relevant perfect number, but differs by some order 2^k. Hence a k-Almost Perfect Number is such a number differing by 2^k. A 0-Almost Perfect Number is of course trivially perfect. A 'Perfect number' is defined as the product of a Mersenne prime and the power of two that is roughly half its prime magnitude : N = (2^n-1)\cdot 2^{n-1} :where (2^n-1) is a Mersenne prime :PN = (MP)*LP2 (Pefect number = Mersenne prime multiplied by the Last Power of 2 it exceeds in value) : for Mersenne primes (3, 7, 31, 127, 8191, ...) :PN = 3*2, 7*4, 31*16, 127*64, 8191*4096, ... :PN = 6, 28, 496, 8,128, 33,550,336, ... Following the same construction for the almost Mersenne primes (aMp), one gets the almost Perfect numbers (aPn): : N = (2^n-1)\cdot 2^{n-1} :where (2^n-1) is an almost Mersenne prime :aPn = (aMp)*LP2 (almost Pefect number = almost Mersenne prime multiplied by the Last Power of 2 it exceeds in value) : for almost Mersenne primes (0, 1, 15, 63, 255, 511, 1023, 2047, 4095, ...) :aPn = 0*½, 1*1, 15*8, 63*32, 255*128, 511*256, 1023*512, 2047*1024, 4095*2048, ... :aPn = 0, 1, 120, 2016, 32640, 130816, 523776, 2096128, 8386560, ... k-almost Perfect numbers Taking the generalisation even further, one can allow for variations of the index of the power of 2 that the Mersenne prime (or 'almost Mersenne Prime'/Mersenne 'almost prime') is multiplied by in forming the perfect number: Two possible formulations are outlined below: Define k-almost Perfectness as a quality of numbers of the form: : N = (2^n-1)\cdot 2^{m-1} : where n and m are non-negative integers It is clear that such numbers are perfect if and only if (2^n-1) is a Mersenne prime AND n is equal to m. If we define k-almost Perfectness as a quality of Mersenne primes when multiplied by powers of 2 OTHER than 2^n-1 then the absolute difference between m and n can define the level of 'almost Perfectness': e.g. : 6 = 3*2 with |m-n|=|2-2|= 0 implying 6 is a 0-almost Perfect number (by definition this is a Perfect number) : 12 = 3*4 with |m-n|=|3-2|= 1 implying 12 is a 1-almost Perfect number : 7 = 7*1 with |m-n|=|1-3|= 2 implying that 7 is a 2-almost Perfect number : 1016 = 127*8 with |4-7|= 3 implying that 1016 is a 3-almost Perfect number :note: all aPn's in this definition can be expressed as a Perfect number multiplied by some power of 2, allowing almost perfectness to be expressed as k = |log_2(aPn/PN)| where aPn/PN gives the ratio of the aPn to the Perfect Number of which it is a multiple/fraction. (e.g. 1984 = 496*4 with |log_2(aPn/PN)|=|log_2(4)|= 2 implying that 1984 is a 2-almost Perfect number k-almost almost Perfect numbers (k-aaPn's) Alternately, one can include 'almost Mersenne primes' and generate k-almost almost Perfect numbers (k-aaPn's) whose values can not all be expressed as a Perfect number multiplied by a power of 2. Examples that cannot be expressed under the first definition but can be here would be multiples of the 0-almost Perfect numbers below, who in this definition would be 0-almost almost Perfect yet still not Perfect, unlike above where the 0-perfect numbers were defined as Perfect - i.e. n=m does not guarantee Perfectness if 2^n-1 is not a Mersenne prime: : 0 = 0*½ with |m-n|=|0-0|= 0 implying 0 is a 0-almost almost Perfect number (yet not a Perfect Number) : 1 = 1*1 with |m-n|=|1-1|= 0 implying 0 is a 0-almost almost Perfect number (yet not a Perfect Number) : 120 = 15*8 with |m-n|=|4-4|= 0 implying 0 is a 0-almost almost Perfect number (yet not a Perfect Number) : 2,016 = 63*32 with |m-n|=|6-6|= 0 implying 0 is a 0-almost almost Perfect number (yet not a Perfect Number) : 32,640 = 255*128 with |m-n|=|8-8|= 0 implying 0 is a 0-almost almost Perfect number (yet not a Perfect Number) : 130,816 = 511*256 with |m-n|=|9-9|= 0 implying 0 is a 0-almost almost Perfect number (yet not a Perfect Number) : 523,776 = 1023*512 with |m-n|=|10-10|= 0 implying 0 is a 0-almost almost Perfect number (yet not a Perfect Number) : 2,096,128 = 2047*1024 with |m-n|=|11-11|= 0 implying 0 is a 0-almost almost Perfect number (yet not a Perfect Number) : 8,386,560 = 4095*2048 with |m-n|=|12-12|= 0 implying 0 is a 0-almost almost Perfect number (yet not a Perfect Number) Mersenne primes as 'almost Perfect numbers' Defining almost-perfectness by magnitudes of 2 that separate a number from its Perfect number multiple/fraction allows a clear understanding of the Mersenne primes as 'almost perfect numbers' which 'become perfect' under multiplication by the correct power of 2: :3 = 3*1 with |m-n|=|1-2|= 1 implying 3 is a 1-almost Perfect number :7 = 7*1 with |m-n|=|1-3|= 2 implying 7 is a 2-almost Perfect number :31 = 31*1 with |m-n|=|1-5|= 4 implying 31 is a 4-almost Perfect number :127 = 127*1 with |m-n|=|1-7|= 6 implying 127 is a 6-almost Perfect number :etc.. Links to Other Almost-Perfect Numbers :"An almost perfect number, also known as a least deficient or slightly defective (Singh 1997) number, is a positive integer n for which the divisor function satisfies \sigma(n)=2n-1 . The only known almost perfect numbers are the powers of 2, namely 1, 2, 4, 8, 16, 32, ... (OEIS A000079)." :"A quasiperfect number, called a "slightly excessive number" by Singh (1997), is a "least" abundant number, i.e., one such that \sigma(n)=2n+1. :Quasiperfect numbers are therefore the sum of their nontrivial divisors. No quasiperfect numbers are known, although if any exist, they must be greater than 10^35 and have seven or more distinct prime factors (Hagis and Cohen 1982)." :"In number theory, an Erdős–Nicolas number is a number that is not perfect, but that equals one of the partial sums of its divisors. That is, a number n is Erdős–Nicolas number when there exists another number m such that : \sum_{d\mid n,\ d\leq m}d=n. " :"In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number n such that the sum of all divisors of n (the sum-of-divisors function σ(n)) is equal to 2n − 1, the sum of all proper divisors of n, s(n) = σ(n) − n, then being equal to n − 1. " :"In number theory, a Descartes number is an odd number which would have been an odd perfect number, if one of its composite factors were prime. They are named after René Descartes who observed that the number D = 32⋅72⋅112⋅132⋅22021 = (3⋅1001)2⋅(22⋅1001 − 1) = 198585576189 would be an odd perfect number if only 22021 were a prime number" :"In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. :For a given natural number k, a number n is called k-perfect (or k-fold perfect) if and only if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number." :"In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number." ---- }} Category:Numbers Category:Number Theory Category:Mathematics Category:Almost Perfect Numbers